22/2/26
Recap: Variation of Parameters
- , where are solutions of the homogeneous differential equation (DE).
-
Theorem
Let be linearly independent (LI) solutions of , where and are continuous functions of on , and let .
Then,
Proof
We have,
By the row operation :
Existence and Uniqueness of IVP of Order Equation
Theorem
Let be continuous in the domain . Then, any function is a solution of the IVP on if and only if it is a solution of the integral equation
Proof
Let be a solution of the IVP (1).
Since is differentiable, is continuous as well.
is continuous and is continuous is integrable.
is a solution of the integral equation (2).
Note that and .
Lipschitz Condition
Definition
A function defined on a domain is said to satisfy the Lipschitz condition with respect to , if there exists a constant such that
where and .
We also say , where is the Lipschitz constant.
Example.